We tend to think that mirrors reverse left to right, but in fact they reverse “back to front,” along an axis perpendicular to the mirror’s surface. Our confusion arises when we misinterpret this.
“[I]magine a back-front reversal of yourself, with your nose, face, eyes, and so forth pushed through to the back of the head, and your back somehow oozed through to the front,” writes University of Auckland psychologist Michael C. Corballis. “You might then ‘feel’ your watch as having remained on the left wrist (say), while back and front have reversed. However it is likely that you will also experience a strong compulsion to recalibrate your internal axes, and then feel the watch to be on the right wrist. In short, a back-front reversal is reinterpreted as a left-right reversal.”
“In his book Amusements in Mathematics, H.E. Dudeney presents a method of classifying 4×4 magic squares based on the distribution of their 8 complementary pairs 1 & 16, 2 & 15, .., 8 & 9. There are just 12 distinct such distributions or ‘graphic types’, which he labelled I to XII. The square above is an example of a type X square.”
The sum of the first n square numbers is n(n+1)(2n+1)/6.
These sums comprise the square pyramidal numbers — each corresponds to the number of oranges that can be stacked in a square pyramid whose base has side n.
This visual proof, for n=3, shows that six square pyramids with n steps fit in a cuboid of size n(n + 1)(2n + 1).
The sea urchin Coelopleurus exquisitus was discovered on eBay. Marine biologist Simon Coppard was directed to a listing on the site in 2004 and realized that the species had not previously been described. When it was properly named and introduced in Zootaxa two years later, the value of specimens on eBay shot up from $8 to $138.
In 2008 a fossilized aphid on eBay was similarly found to be unidentified. Eventually it was named Mindarus harringtoni, after the buyer.
The dark polyomino at the center of this figure, devised by Craig S. Kaplan, has an unusual property: It can be surrounded snugly with copies of itself, leaving no overlaps or gaps. In this case, the “corona” (red) can be surrounded with a second corona (amber), itself also composed of copies of the initial shape. But that’s as far as we can get — there’s no way to create a third corona using the same shape.
That gives the initial shape a “Heesch number” of 2 — the designation is named for German geometer Heinrich Heesch, who had proposed this line of study in 1968.
Shapes needn’t be polyominos: Heesch himself devised the example below, the union of a square, an equilateral triangle, and a 30-60-90 triangle:
It earns a Heesch number of 1, as it can bear only the single corona shown.
Can all positive integers be Heesch numbers? That’s unknown. The Heesch number of the square is infinite, and that of the circle is zero. The highest finite number reached so far is 6.
A train line extends infinitely far east and west. Stations are spaced a mile apart, and midway between each pair of stations is a signal light. There’s one train on the line, and it moves to an adjacent station at the top of each hour. Its choice (east or west) is determined by the engineer, who flips a coin 20 minutes after each hour. If the coin lands heads, the train’s next destination will be the station one mile east. If it lands tails, the train will next go to the station one mile west.
Forty minutes after each hour, one of the infinitely many signal lights will flash. The flash is visible all along the line. The identity of the flashing light is random, and it’s unrelated to the coin toss.
You wake up on this train while it’s stopped at a station. It’s 2:30 p.m., which means the engineer flipped his coin 10 minutes ago. The conductor tells you that the next destination is Willoughby. A map tells you that the Willoughby station is flanked by East Willoughby and West Willoughby, so you must be at one of these two stations, but you don’t know which one. Because these are the only two possibilities and there’s no reason for one to be more likely, you conclude that they’re equally probable.
Before the train departs at 3 p.m., is it possible to guess the outcome of the engineer’s coin toss at 2:20 p.m. with success probability greater than 1/2? El Camino College mathematician Leonard M. Wapner contends that it is. At 2:40 p.m. a signal light will flash. There’s a nonzero probability p that the light that flashes will be one of the two lights adjacent to the Willoughby station. If that happens, it will indicate with certainty the direction of Willoughby (east or west) from your current location. The chance that the flash doesn’t come from one of these two lights is 1 – p, and in that case the chance is 1/2 that it comes from the direction of Willoughby. Overall:
“So,” Wapner writes, “if the light flashes to your east, you would guess that the train will be departing to the east and that the engineer’s coin landed heads. If the light flashes to the west, you would guess that the train will depart to the west and that the engineer’s coin landed tails. You should expect your guess (east/heads or tails/west) to be successful more often than not.”
(He adds, “The Willoughby prediction scheme, though mathematically valid, is far too contrived for it to be achieved in actuality. But there being no mathematical contradictions, the door remains open to variations and applications.”)
